bubbles and crashes with partially sophisticated...

Bubbles and Crashes with Partially

Sophisticated Investors�

Milo Bianchiy Philippe Jehielz

December 1, 2008

Abstract

We consider a purely speculative market with �nite horizon and

complete information. We introduce partially sophisticated investors,

who know the true average buy and sell strategies of other traders,

but lack a precise understanding of how these strategies depend on

the history of trade. In this setting, it is common knowledge that the

market is overvalued and bound to crash, but agents hold di¤erent

expectations about the date of the crash. We de�ne conditions for

the existence of equilibrium bubbles and crashes, characterize their

structure, and investigate whether bubbles may last longer when the

amount of fully rational traders increases.

Keywords: Speculative bubbles, crashes, bounded rationality.

JEL codes: D84, G12, C72.

�We thank numerous seminar participants and in particular Cedric Argenton, AbhijitBanerjee, Markus Brunnermeier, Mike Burkart, Sylvain Chassang, Gabrielle Demange,Daniel Dorn, Tore Ellingsen, Botond Koszegi, John Moore, Marco Ottaviani, Peter Rau-pach, Marcus Salomonsson, Johan Walden, Jörgen Weibull, Muhamet Yildiz and RobertÖstling for useful discussions. Milo Bianchi gratefully acknowledges �nancial support fromRegion Ile-de-France and from the Knut and Alice Wallenberg Foundation.

yParis School of Economics. E-mail: [email protected] School of Economics and University College London. E-mail: [email protected]

1

1 Introduction

In a speculative bubble, trade occurs at prices above fundamentals only be-

cause investors expect the selling price to be even higher in the near future

(Stiglitz, 1990). Trading activities based on speculative motives seem wide-

spread and have been widely documented (see Garber, 1990; Kindleberger,

2005), but their foundation remains largely unclear. As purely speculative

trade must rely on some inconsistent expectations or suboptimal decisions,

it cannot arise in standard rational expectation models (Tirole, 1982).

In this paper, we show that speculative bubbles may be explained by

considering that a fraction of investors have a partial rather than total un-

derstanding of the investment strategies employed by other investors. Specif-

ically, in our model, all investors understand the aggregate buying and selling

pressures that apply on average throughout the entire duration of the mar-

ket. But, while some investors also understand how these buying and selling

pressures precisely depend on the history of trade (these are fully rational

traders), others lack such precise knowledge (these are partially sophisticated

traders). As a result, some investors form erroneous beliefs about the date

of the crash, and they are not able to sell just before it, thereby allowing

bubbles and crashes to occur.

Before we present in more detail the features of our model and their

implications, we make two observations about bubbles, which will also allow

us to explain how our approach di¤ers from other approaches.1

The �rst observation is that assuming traders have private information

is neither necessary nor su¢ cient for the emergence of bubbles. From a

theory viewpoint, private information alone cannot explain bubbles, as can

be inferred from the no-trade theorems (Milgrom and Stokey, 1982). From

an experimental viewpoint, there is substantial evidence that bubbles emerge

even in contexts in which by design the structure of the game and the value

of future dividends are commonly known to subjects (see Porter and Smith,

2003, for a review). Such an observation leads us to consider a model with

complete information in which the structure of fundamentals is commonly

1More precise references to the literature on bubbles are provided in Section 5.1.

2

known but traders are heterogeneous in their ability to understand others�

trading strategies.

The second observation is that, in real bubble episodes, it seems highly

plausible that most agents realize, at least eventually, to be in a speculative

market. There are anecdotes about this2, and somewhat systematic survey

evidence. Shiller (1989), for example, reports that just before the U.S. stock

market crash of October 1987, 84% of institutional investors thought that

the market was overpriced; 78% of them thought that this belief was shared

by the rest of investors and, still, 93% of them were net buyers. Such an

observation calls in our view for a new modelling of bounded rationality, al-

lowing (at least a fraction of) agents to have some but not full understanding

of the situation. In most existing approaches, agents are assumed to be ei-

ther fully rational, thereby, in equilibrium, having a complete understanding

of the market dynamics, or completely mechanical, and so lacking any un-

derstanding that they are in a bubble and that the market may crash. By

contrast, in our model, it is common knowledge among agents that they are

in a bubble and that the market must crash. Bubbles emerge as all agents

believe, rightly (when rational) or wrongly (when partially sophisticated),

that they can pro�t by investing in the speculative market and exiting at the

right time.

We now describe in some more detail our framework. We consider a mar-

ket in which agents can trade an asset with no fundamental value. Trade

can occur only for a �nite number of periods, as liquidity shocks force a

(small) fraction of agents to leave the market in each period. Investors are

either fully rational or partially sophisticated. Fully rational investors have

rational expectations, as usual. Partially sophisticated investors understand

the aggregate buy and sell rates along the duration of the speculative mar-

ket, without having a precise perception of how these rates vary along the

life-cycle of the bubble. Moreover, they adopt the simplest theory of trade

2For example, Eric Janszen, a leading commentator of speculative phenomena, wrotean article in the middle of the Internet bubble (November 1999) saying: "During the �nalstages, the mania participants �nally admit that they are in a mania. But they rationalizethat it�s OK because they � only they and not the other participants �will get out intime." (article accessible at www.bankrate.com)

3

volumes and price dynamics that is compatible with their knowledge, thereby

expecting constant buy and sell rates throughout the duration of the spec-

ulative market, independently from the history of trades.3 In equilibrium,

these constant rates match the aggregate intensities averaged over time, as

resulting from the actual sell and buy strategies. In each period, based on

their expectations, agents make the optimal trading decisions.

Before previewing our main results, let us emphasize that this equilibrium

is not, in our view, the result of individual learning, whereby the same group

of investors repeatedly acts in the same market, since in this case partially

sophisticated investors would probably either learn that they get exploited or

they would run out of money. We think instead of a process of learning at an

historical level, in which a new cohort of investors enters the market in each

bubble episode. These investors interpret the current market in light of some

historical data about similar episodes. Fully rational investors analyze these

data with elaborate statistics and as a result they rightly understand the

trade dynamics. Partially sophisticated investors, instead, use a simpli�ed

model, able to provide the correct averages but no more detailed statistics.4

In a sense, they apply a linear model to analyze trade dynamics that are not

necessarily linear (as in the spirit of Sargent, 1993).

As it turns out, in our model, optimal investment strategies depend only

on investors�expectations about the market dynamics one period ahead. In

line with the experimental evidence reported by Haruvy, Lahav and Noussair

(2007), these expectations can be described by a simple function of investors�

knowledge about past bubble episodes and of the behaviors observed during

the current bubble episode.

Such an interaction between historical knowledge and current trends gen-

erates a rich pattern of expectations and behaviors. For partially sophis-

3This assumption may be viewed as being in line with the observation that the dateof the crash tends to appear quite similar to many other days. Investors and analyststypically �nd it hard to understand in what sense that precise day was so special, andfundamentally di¤erent from the previous day. Even the systematic analysis by Cutler,Poterba and Summers (1989) concludes that "many of the largest market movements inrecent years have occurred on days when there were no major news events."

4This may be so because these averages may be easier to understand and rememberthan more detailed information about say the daily buy and sell rates.

4

ticated investors, historical knowledge is based on the average strategy ob-

served along a complete bubble episode, which include days in which most

people want to buy and days in which most people want to sell. As a result,

these investors get a positive surprise after any "good day" and a negative

surprise after any "bad day". For these investors, then, a series of good days

leads to euphoria and a series of bad days (or one very bad day) leads to

panic.

This mechanism builds the foundation of bubbles and crashes in our

framework. Along the bubble equilibrium, investors �rst observe a series

of rising prices, due to excess demand for speculative stocks. Partially so-

phisticated investors interpret such an increase in prices as a sign that the

bubble will last longer.5 Hence, they decide to remain invested longer than

they had planned, and, in doing so, they end up overestimating the duration

of the bubble. These behaviors can be exploited by fully rational investors,

who feed the bubble for a while and exit just before the endogenous crash.

Upon observing the massive sale by rational investors, partially sophisticated

investors realize it is time to sell (actually, it is too late for most of them),and this indeed leads to the crash.6 In this way, our framework generates

both bubbles and crashes, phenomena which tend to be considered separately

in the literature.7

Inspired by the e¢ cient market hypothesis, whereby rational investors

play a stabilizing role and ensure market e¢ ciency, we then explore the rela-

tion between bubbles and the share of rational investors in our setting. We

observe that rational investors should be neither too many nor too few for

bubble equilibria to arise with the property that, just before the crash, there

is a panic phase in which investors realize everyone is trying to sell and the

crash is about to occur.8 We also observe in our basic model that when there5This captures a strong regularity documented in Shiller (2000). As the price increases,

more people display "bubble expectations", i.e. the belief that, despite the market beingovervalued, it will still increase for a while before the crash.

6Kindleberger (2005) provide a rich historical account of such periods of euphoria andpanic; Brunnermeier and Nagel (2004) and Temin and Voth (2004) document how majorinvestors earn great pro�ts by rightly timing the market.

7See Section 5.1 for elaboration on this.8With too many rational agents, bubbles cannot arise because we are too close to a

5

are more rational investors the maximal duration of a bubble gets smaller.

However, by extending our analysis to a setting with uncertainty aversion,

we show that bubbles may last longer as the fraction of rational investors

increases.9 Thus, in our setting, whether rational investors have a stabilizing

role depends on the attitude of investors toward uncertainty.

The rest of the paper is organized as follows. In Section 2 we describe the

model and the solution concept. In Section 3 we analyze bubble equilibria,

characterizing the conditions for their existence and showing how the max-

imal duration of bubbles varies as a function of our parameters. In Section

4 we explore whether rational agents have a stabilizing role. In Section 5 we

discuss some related literature, policy implications, and avenues for future

research. Omitted proofs are provided in the Appendix.

2 The model

Our economy is populated by a unit mass of risk neutral individuals.10 Ini-

tially, a fraction K of them is endowed with cash, each owning w > 0; and

a fraction (1 � K) is endowed with stocks, each owning one stock. Cashand stocks are distributed independently across agents. The value of cash

is constant over time. Stocks are purely speculative: they pay no dividend,

their fundamental value is zero, and their return is given only by changes in

the price pt. For simplicity, we assume that each agent can hold at most one

stock at a time, and each stock is indivisible.11

rational expectation model. With too few rational agents, boundedly rational traders donot observe any massive sale before the crash occurs and as a result they do not get tounderstand that the crash is about to occur before it actually does.

9In fact, facing less strategic uncertainty, rational agents are more prone to invest thanboundedly rational ones. As the share of rational investors increases, more people enter thespeculative market, which may induce boundedly rational agents to be more optimistic,thereby allowing to sustain longer bubbles.10Section 4.3 considers the case of uncertainty averse agents.11The substance of our analysis would not change if stocks were perfectly divisible and

everyone could spend his entire wealth in stocks. The crucial assumption, as we shall see,is that the entire economy has �nite wealth.

6

2.1 Financial market

In each period t = 1; 2; :::, individuals can trade. Within each period t, trad-

ing occurs as follows: individuals decide simultaneously whether to submit

their orders, a market clearing price pt is announced, and orders are cleared.

Borrowing stocks or cash is not allowed, so the investment option for

individual i in period t is simply fbuy; stay outg if i holds cash at t; or fsell;stay ing if i holds a stock at t. In addition, in case he submits a trade order,each individual speci�es a maximum price pbt at which he is willing to buy

or a minimum price pst at which he is willing to sell. Such reserve prices will

ensure that demand and supply are smooth functions of the price.

While individual decisions on whether or not to submit a trade order are

endogenously determined in equilibrium, for simplicity we leave reservation

prices exogenous. For each individual, these prices write as pst = �pt�1 and

pbt = �pt�1+(1��)w; where the parameter � is drawn independently acrossindividuals from a commonly known distribution with smooth density and

support on [0; 1].12 The induced distributions of reservation prices for those

agents who may sell or buy a stock in period t are described by the cumu-

lative functions �t and �t; respectively. These distributions depend on the

history of trades, but from the above assumption their support always lies

respectively in [0; pt�1] and in [pt�1; w].

The market clearing price pt can be characterized in every period t as fol-

lows. Denote the amount of buy and sell orders at t byBt and St, respectively:

These quantities can be written as

Bt = �tKt and St = �t(1�K); (1)

where Kt denotes the amount of people who can buy a stock, �t the share of

those who want to buy, and �t the share of those who want to sell in period

t.13 If Bt � St; the price pt solves Bt[1��t(pt)] = St; which implies �t(pt) � 012A fully speci�ed model may derive such � from heterogeneous preferences, regarding

for example attitudes towards uncertainty.13These rates are in principle the realization of a random variable that aggregates the

strategies of every agent at a given point in time. However, since we consider a setting witha continuum of agents, each realization of such variable corresponds to its expected value.

7

and so pt � pt�1: If instead Bt < St; the price pt solves Bt = St[1 � �t(pt)];which implies �t(pt) > 0 and so pt < pt�1: Hence, we have

pt � pt�1 , Bt � St: (2)

Finally, we assume that, at the end of each period t, each agent observes the

trading price pt and the volume of trade Vt = minfBt; Stg, from which he

can correctly infer Bt and St: In what follows, we refer to Bt and St simply

as demand and supply in period t.

2.2 Exit from the market

While the amount of stocks is �xed to (1�K) throughout the analysis, theamount of people who can buy stocks changes over time due to the exit of

some investors. We de�ne exit from the speculative market in period t as

the sum of stock-holders who sell and cash-holders who decide not to buy in

period t. That is, the amount of exit in period t is

Et = Vt + (1� �t)Kt: (3)

Selling may either be deliberate or it may be induced by a liquidity shock,

which forces some agents to sell the stock immediately and stay out of the

market from then on.14 An agent with stock may be hit by a shock with

probability z > 0; thus the amount of exogenous exit in each period is z(1�K), where we assume that z(1�K) < K:15

In equilibrium, agents who decide to exit the speculative market never

wish to re-enter, and we assume that this is rightly understood by every-

body.16 Accordingly, the amount of people who can buy a stock at t evolves

Accordingly, in what follows, we simplify the notation and ignore the distinction betweenthe expected values of these quantities in a given period and their actual realizations.14Such shocks should not be confused with noise trade. They are simply to avoid the

possibility that the speculative market lasts forever.15Assuming, perhaps more naturally, that everyone may be hit by a liquidity shock

would not change our results, but it would complicate their derivation.16In Bianchi and Jehiel (2008), we show that, under a (natural) assumption, this is the

only consistent case.

8

as

Kt+1 = Kt � Et; (4)

and, by equation (3), we have

Kt+1 = �tKt � Vt: (5)

From equation (5), it follows that the price pt never recovers after having

dropped. If in period t the price drops, it must be due to excess supply in

t, in which case the volume of trade Vt is equal to the demand �tKt and

equation (5) yields Kt+1 = 0. By equation (4), Kt can only decrease over

time, which implies that Kt+s = 0 for all s � 1: Thus, after a price drop, themarket closes.

2.3 Cognitive abilities and equilibrium

Agents di¤er in their ability to understand other agents�trading strategies.

An agent�s type � determines his expectation about the dynamics of trade

volumes and the associated prices. Such dynamics depend on demand and

supply in each period, which in turn depend on the amount of agents still

active in the market together with their buy and sell strategies.

From de�nition (1), the period s expectations for an agent of type � about

the demand and supply in period t are given by

B�;st = ��;st K�;st and S�;st = ��;st (1�K) for every s � t; (6)

where ��;st and ��;st are this agent�s expected buy and sell rates, and K�;st

is the expected amount of potential buyers in the market at t: In order to

estimate the latter, agents need to know how many traders are in the market

at s; and how many exit from s to t � 1. Recall that, in period s, agentshave observed the history of prices and trade volumes, by which they can

correctly infer the amount of exits until s� 1 and so Ks: Hence,

K�;ss = Ks; for every � and s.

9

For t > s; using equation (4), we have

K�;st = Ks �

w=t�1Xw=s

E�;sw ; (7)

where, by equation (3),

E�;sw = min(B�;sw ; S�;sw ) + (1� ��;sw )K�;s

w : (8)

Given equations (6), (7) and (8), an agent�s expectation about future

market dynamics are completely characterized by his expectations about fu-

ture buy and sell rates. These expectations depend on the agent�s type, as we

now describe. For simplicity, we consider a setting with only two cognitive

types: standard rational agents R and partially sophisticated agents I, in

proportion r and (1� r), respectively.R-types understand perfectly well the patterns of other investors�strate-

gies. Hence, if the actual buy and sell rates arising in equilibrium in period

t are given by �t and �t, R-agents�expectations must satisfy

�R;st = �t and �R;st = �t for every s � t: (9)

I-agents, instead, expect constant buy and sell rates throughout the duration

of the speculative market, where these rates coincide with the actual average

rates that prevail throughout the speculative market. Formally, denote with

T + 1 the last date in which the speculative market operates, as determined

endogenously in equilibrium. The average buy rate �� and the average sell

rate �� for the sequence of buy and sell decisions arising in equilibrium are

�� =1

T + 1

T+1Xt=1

�t and �� =1

T + 1

T+1Xt=1

�t: (10)

I-types�expectations are required to correspond to such averages, hence we

have

�I;st = �� and �I;st = �� for every s � t: (11)

10

We �rst note that I-agents hold �I;st and �I;st constant irrespective of the

histories of trade. For example, they may think of these rates as resulting

from a given distribution of strategies, of which they know only the mean.

In this way, even when observing a realization di¤erent from the mean, they

need not change their theory about the underlying distribution.17

Investment decisions are determined as the optimal investment strategies

given these expectations and agents�payo¤s, which are de�ned as follows.

The payo¤ of an agent is zero if he holds cash or stock forever; (ps� pt) if hebuys a stock at time t and he sells it at time s; ps if he initially owns a stock

and sells it at s; and �pt if he buys a stock at t and keeps it forever.18 Aftereach history of prices and trade volumes, each agent chooses an investment

strategy that maximizes his expected payo¤. An investment strategy pro�le

speci�es an investment strategy for every agent in the economy, which serves

to de�ne an equilibrium in our setting.

De�nition 1 (Equilibrium): An investment strategy pro�le is an equilib-rium if, all along the equilibrium path, each agent�s investment strategy max-

imizes his expected payo¤, given the expectations de�ned in equations (9) and

(11).

Our de�nition of equilibrium is in the spirit of the rational expectation

equilibrium in which, due to the dynamic nature of the interaction, beliefs

and investment strategies must be optimally adjusted at every point in time.

Note however that our de�nition only considers the incentives of agents on

the equilibrium path, and not the adjustment of beliefs and strategies after

a positive mass of agents have made non-equilibrium decisions.19

17We also prefer having in mind that I-agents are not aware that other investors mayhave a more accurate understanding of the market dynamics. Otherwise, given that inour model trades mostly occur for speculative reasons, I-agents may simply decide to stayoutside the market if they realize to be less sophisticated than others. Such a conclusionwould be altered if all agents thought that some other agents are less sophisticated thanthey are. We leave the extension of our model to the case of in�nitely many cognitivetypes (that would allow for this) for future research.18For an agent hit by a liquidity shock, payo¤s may be described di¤erently. For example,

such agent may only care about immediate cash, and place no value on cash in the future.19While one could easily amend the solution concept to cover o¤-the-path optimizations

11

3 Analysis

3.1 Optimal investment strategies

We focus on symmetric equilibria in pure strategies, where all investors of

a given type and with a given endowment in period t follow the same pure

strategy. Observe �rst that the existence of an equilibrium is not an issue,

as there is always the non-bubble equilibrium in which every agent exits the

speculative market at the very �rst period.20 Our interest lies in showing the

possibility of bubble equilibria, and characterizing the conditions for such

equilibria to exist. Given that the fundamental value of the asset is zero, we

de�ne any situation in which trade occurs as a bubble. Conversely, if at some

point no one is willing to buy the stock at any price, the speculative market

closes. Provided that some trade had occurred, we then say that there is a

crash.

The problem faced by an individual of a given type is the same irrespective

of whether he has cash or stock. That is, for any agent i 2 � with cash andany agent j 2 � with stock (who is not hit by a liquidity shock), i prefers tobuy if and only if j prefers to stay in, and i prefers to stay out if and only if

j prefers to sell. Intuitively, trade occurs either among people with di¤erent

needs, as described by the liquidity sellers z, or among those with di¤erent

expectations, as described by the di¤erent types.

In principle, investment strategies may be very complicated, since each

agent may condition his current strategy on the whole history of past trades

and on his own past trading decisions. However, as it turns out, our model

allows a very simple representation of optimal trading strategies. We start

by showing that, expecting all exits from the market to be permanent, no

agent wishes to re-enter after having exited, and so equation (4) is indeed

and expectations, this would make the notation heavier (in particular, the state variableparameterizing the decisions should no longer be the calendar time t but the entire his-tory of buy/sell decisions) without adding much economic insight. Moreover, since eachindividual agent has a negligible weight (there is a continuum of agents), our notion ofequilibrium is in the spirit of the Nash equilibrium, where no single agent can on his ownmove the system away from the equilibrium path.20In this equilibrium, I-agents� expectations are correct, and their decisions to exit

immediately is thus rational.

12

consistent with equilibrium behaviors.

Lemma 1 Expecting exits to be permanent, an agent who exits the specula-tive market at t prefers to stay out from then on.

Proof. See Section 6.1 in the Appendix.

As noticed after equation (5), the fact that exits are permanent implies

that the price never recovers after having dropped. As a result, optimal

trading strategies at time t can be expressed as a function of the expected

prices at t and at t+ 1 only. We state the result in the next Proposition.

Proposition 1 An agent i 2 � prefers to buy/stay in the market at t if andonly if

p�;tt+1 � p�;tt ; (12)

or, equivalently, if and only if

B�;tt+1 � S�;tt+1: (13)


Proposition 1 allows us to write the optimal investment strategy for I-

agents in any period t simply as a function of the observed amount of people

who can buy at t and of the constant expectation about future buy and sell

rates. We express this more precisely in the next Corollary.

Corollary 1 An agent i 2 I prefers to buy/stay in the market at t if andonly if

Kt � W; (14)

where

W � ��(1�K)(1 + ��)��2

: (15)


In sum, at any t, optimal trading strategies for I-agents are only a function

of Kt; which describes the history of trades, and of the expectation about

13

future buy and sell rates, as expressed in equation (15). Such expectation,

in turn, depends on the equilibrium duration of the bubble; but it remains

constant over time. R-agents�expectation, instead, re�ects the true strategies

observed along the equilibrium, and so it may vary with t: In particular, given

equation (13), these agents buy/stay in the market in period t if and only if

�t+1Kt+1 � �t+1(1�K):

3.2 Bubble equilibria

We can now show that, under conditions to be characterized in the next

Subsection, there exist equilibria of the following form. Apart from a share

z of stock-holders who sell in each period due to liquidity shocks, investors�

strategies are such that in each period t � T � 1 everyone tries to enter thespeculative market and no one wants to sell; in period T , I-investors buy and

R-investors sell; at T + 1; everyone tries to sell but no one is willing to buy.

The crash then occurs and the market closes. Along these equilibria, called

bubble equilibria, the aggregate buy and sell rates are

�t =

8><>:1 for t � T � 1;

1� r for t = T;

0 for t = T + 1;

(16)

and

�t =

8><>:z for t � T � 1;

z + r(1� z) for t = T;

1 for t = T + 1:

(17)

According to equation (10), I-agents are induced to expect the following buy

and sell rates:�� =

T � rT + 1

; (18)

and

�� =Tz + r(1� z) + 1

T + 1: (19)

This allows us to de�ne B�;ts and S�;ts according to equations (6), (7), and

(8). Besides, given the above speci�cations, the only variable remaining to

14

endogenize is the duration T of the bubble. A major objective of the next

analysis is then to characterize the conditions for the existence of such T;

and to understand how T depends on our exogenous parameters K; z and r:

Given that the exits from the speculative market are permanent, if i wants

to sell/stay out at t; then he wants to sell/stay out for every s > t: Likewise,

if i wants to buy/stay in the market at t; this reveals that he wanted to

buy/stay in the market at each s < t: Hence, the equilibrium T is de�ned

simply by three conditions. First, each agent i 2 R has to prefer to buy atT �1; so we must have pT > pT�1: By equation (13), this condition writes as

BT � ST : (20)

Second, each agent i 2 I has to prefer to buy at T; which, by equation (14),writes as

KT � W: (21)

Third, each agent i 2 I has to prefer to sell at T + 1; which, again usingequation (14), writes as

KT+1 < W: (22)

The last two conditions also imply that each agent i 2 R prefers to sell at T;since given I-agents�behavior, the market crashes at T + 1: We summarize

these observations in the following Proposition.

Proposition 2 Any T satisfying conditions (20), (21) and (22) can be sus-tained as a bubble equilibrium.

3.2.1 Example

While we postpone a more detailed analysis of conditions (20), (21) and

(22) to Section 6.4 in the Appendix, we now highlight their structure with a

numerical example. Suppose that z = 0:2; r = 0:2 and K = 0:9.

In Figure 1, the solid curve is the function W (T ), the solid line is the

function F (T ) = K � z(1 � K)(T � 1), the dashed line is the functionG(T ) = (1 � r)[K � z(1 � K)T ] � r(1 � K). F (T ) and G(T ) map the

15

10 20 30 40 500.0

0.2

0.4

0.6

0.8

T

T1

W(T)

F(T)

G(T)

T2T3

Figure 1: Conditions de�ning the equilibrium bubble for z=0.2, r=0.2 and K=0.9.

equilibrium T with the amount of investors who can buy in period T and

T + 1, respectively. These functions are derived in Section 6.4, and, by

construction, they are such that F (T ) = KT and G(T ) = KT+1. The vertical

line plots T = T1; as derived from condition (20). In this example, condition

(20) is satis�ed for T � T1; condition (21) for T � T2; as de�ned by the

intersection of W (T ) and F (T ); and condition (22) for T > T3; as de�ned by

the intersection of W (T ) and G(T ): Speci�cally, substituting our values in

equations (20), (21) and (22) we �nd that, up to integer approximations, they

require respectively T � 42; T � 43; and T � 41: Hence, any T 2 [41; 42]can be a bubble equilibrium.

We now describe how investors� expectations evolve along the bubble

equilibrium in this example. Consider the equilibrium in which all rational

agents sell at T = 42. Fully rational agents have rational expectations, and

so they expect the crash to occur at t = 43 throughout the duration of

the bubble. Instead, as mentioned in Introduction, I-agents�expectations

may change once they observe the actual buy and sell rates realized in each

period. On the one hand, after any t < T; I-agents revise their expectation

16

Figure 2: Equilibrium strategies and I-agents�expectations for z=0.2, r=0.2 and K=0.9.

about the date of the crash upwards. This is so because, in all these periods,

�t > �� and �t < ��. On the other hand, at date T , I-agents revise their

expectation about the date of the crash downwards (provided that r is not

too small, see Section 4.2 for details). This is so because �T < �� and �T > ��.

These patterns are shown in Figure 2. The increasing solid line describes

the actual supply observed along the equilibrium St (as determined by equa-

tion (17)). This induces I-agents�expected supply SI;st = ��(1�K) for eacht and s (as determined by equation (19)). This expectation is represented by

the dashed constant line SI . The decreasing solid line describes the actual

demand Bt (as determined by equation (16)), which determines I-agents�

expected demand according to BI;st = ��KI;st for each t � s (as determined

by equations (7), (8) and (18)). The decreasing dashed lines BI;s represent

the period s expectations for I-agents about the demand in all periods t � s.The expected date of the crash in each period s � T; which is de�ned by

mint

nt j BI;st < SI;st

o; evolves accordingly: In this example, before any trade

takes place, I-agents expect the crash to occur at t = 27: At the beginning

of the second period, upon having observed fewer exits than expected, they

17

expect the crash at t = 28: Similar dynamics occur at each s � 42 : for

example, at s = 10 they expect the crash at t = 32; at s = 25 they expect

the crash at t = 39; at s = 42, when they buy for the last time, they expect

the crash at t = 45. Only at the end of period 42; upon observing the massive

exit by rational agents (and when it is too late to sell), they realize that the

crash will indeed occur at t = 43.

3.3 Existence and maximal duration of a bubble equi-

librium

We now ask ourselves when a T � 1 satisfying conditions (20), (21) and (22)exists as a function of the parameters K, z and r: When such T exists, we

say that a bubble equilibrium exists. Intuitively, a bubble is more likely to

develop when there is a large amount of cash that could potentially be used

to fuel it; when not too many people are hit by shocks that force them to exit

the speculative market, and when the number of investors who can correctly

predict the date of the crash is not too large. We express this in the following

Proposition.

Proposition 3 There exists a K�(r; z) < 1 such that if K � K�(r; z); then

a bubble equilibrium exists. Such minimal K�(r; z) increases in r and z.


As shown in the previous example, and more generally in Section 6.4,

there need not be only one T satisfying conditions (20), (21) and (22). One

natural point of interest is the largest T that can be sustained in equilibrium,

the one maximizing R-agents�pro�ts.

Such largest T is de�ned by conditions (20) and (21). The �rst condition

can be explained recalling that, even if no one exits voluntarily from the

market, a mass z(1�K) of agents sells in each period due to liquidity shocks.Hence, given that the mass of potential buyers decreases over time, R-agents

cannot exit too late if they want to �nd enough I-agents who buy their

18

stocks. Condition (20) can be written as

T � K � rz(1�K)(1� r) � T1: (23)

Condition (21) instead imposes an upper bound on T whereby, if R-agents

sell too late, I-agents would not buy since the amount of cash observed at

that stage would be too low. Such an upper bound is de�ned by

T � T2;

where T2 is the largest root solving K � z(1�K)(T � 1) = W (see Section

6.4 for details). Accordingly, we de�ne the longest bubble equilibrium as

Tmax � minfT1; T2g:

In order to investigate how Tmax varies with our exogenous parameters, the

�rst issue is under which conditions T1 or T2 is the constraint de�ning Tmax:

As shown in Section 6.6, when the fraction of rational agents r is small, the

latter constraint binds, while the opposite occurs when z or K are small.

Irrespective of this, however, the comparative statics are clear: both T1 and

T2 increase in K and decrease in r and z; as we show in the next Proposition.

Proposition 4 The maximal equilibrium bubble Tmax increases in K and

decreases with z and r: Moreover, Tmax !1 as K ! 1 or z ! 0:


Propositions 3 and 4 show that bubbles are supported by large K, small

z and small r: These relations are consistent with empirical evidence. The

e¤ect of a large K is in line with the observation that speculative stocks tend

to be initially in short supply, and that bubbles are sustained by the large

involvement of new investors (see Cochrane, 2002; Kindleberger, 2005). A

small probability of shock z implies that the fraction of potential investors

decreases slowly, which is consistent with the fact that bubbles tend to display

slow booms and sudden crashes (see Veldkamp, 2005). Finally, the e¤ect of

19

a small r echoes the observation that bubble episodes tend to attract a large

number of inexperienced investors (see Shleifer, 2000; Kindleberger, 2005).

However, as we discuss in the next Section, the relation between bubbles and

rationality is not so clear-cut, once we allow for ambiguity aversion.

4 Bubbles and rationality

In this Section, we discuss further how the existence and the structure of a

bubble equilibrium vary with the share of rational agents in the market.

4.1 Rational investors should not be too many

As in standard models, we cannot have bubbles if all investors are fully

rational. In particular, in a bubble equilibrium, r has to be small enough so

that all rational agents are able to sell at T: This condition de�nes T1; as

expressed in equation (23): As we must have T1 � 1; we need

r � K � z(1�K)1� z(1�K) � rmax:

Hence, we can de�ne a necessary condition for the existence of a bubble

equilibrium.

Proposition 5 If r > rmax; then no bubble equilibrium exists.

4.2 Rational investors should not be too few

As expressed in Proposition 3, bubbles are more likely to arise when the share

of rational investors r is low. On the other hand, in the bubble equilibrium

characterized above, rational investors play a key role. By exiting at T , they

give a negative shock to the market, which makes I-investors aware that they

had overestimated the length of the bubble and that the crash is about to

occur. As a result, I-investors rush to sell as they realize everyone else is

trying to sell. Such �nal panic phase is a rather common feature of market

20

crashes (see Kindleberger, 2005), and we now show that it requires r to be

not too small. Speci�cally, consider the following condition:

BI;T+1T+1 < SI;T+1T+1 ; (24)

which ensures that, at the beginning of T+1; just before the crash occurs, all

investors expect the crash to occur next. Together with condition (21), this

requires that I-investors�expectation about the date of the crash changes

between period T and period T + 1; which in turns requires that some bad

shock occurs in period T: Since the only source of such bad shocks is that

R-investors decide to exit, we need su¢ ciently many of them. We can state

this more precisely with the following Proposition.

Proposition 6 In a bubble equilibrium where I-agents, at the beginning of

T + 1; realize that the market will indeed burst at T + 1, we must have

r > rmin;

where rmin is implicitly de�ned by the condition Trmin = 1:


4.3 Uncertainty Aversion

Proposition 4 shows that bubbles are more likely to last longer when the

fraction of rational investors is smaller. We now show that this need not be

the case if we consider a setting with uncertainty averse agents.21

In our model, uncertainty concerns solely the predictions of what other

investors do. Hence, the amount of uncertainty faced by each agent depends

on his ability to understand other investors�equilibrium strategies. If some

I-agent perceives enough uncertainty, and he prefers to avoid it, he may re-

frain from investing in the speculative market. On the other hand, since

21Uncertainty (or equivalently ambiguity) describes situations in which agents�percep-tions need not be accurate enough to provide them with a unique probability measure overthe possible states of the world.

21

fully rational agents face no uncertainty, they may be more willing to in-

vest. As a result, the amount of investors in the speculative market is in

general increasing with the share of rational agents. This in turn may induce

more optimistic expectations and higher demand, thereby allowing to sustain

longer bubbles.

In order to formally illustrate this idea, we enrich our setting by assum-

ing that, independently from their cognitive types, investors di¤er in their

attitudes towards ambiguity. Such attitudes however are not relevant for

R-investors, since as noted they face no ambiguity. For I-investors, instead,

we distinguish between ambiguity averse investors H and ambiguity neutral

investors L; which have mass (1�r)h and (1�r)(1�h) respectively. Admit-ting that their predictions can be mistaken by some ", H-agents believe that,

in every t, the actual buy rate �t will be in the interval [ �� "; �� + "] \ [0; 1]and the actual sell rate �t will be in the interval [�� "; �� + "]\ [0; 1].22 Fur-thermore, these agents consider the worst realizations of �t and �t; and, given

that, they choose the optimal investment strategy.23 Hence, in order to be

part of speculation, they require a return which compensates the perceived

uncertainty.24 Investors of type L are instead neutral towards uncertainty

(or, alternatively, they do not admit that their predictions can be mistaken).

Hence, as in Section 2, such investors only consider the averages �� and ��:

We here consider the special case of z ! 0 (Bianchi and Jehiel, 2008

provide a more general treatment). In this case, Tmax is de�ned by T1; which

may increase in r since a higher r reduces the mass of ambiguity averse agents

in the market. In fact, the bubble equilibrium is such that H-agents exit at

some ~T , by selling to L- and to R-agents; R-agents sell to L-agents in period

22The error term " is here taken as given. One could for example endogenize this intervalby letting the expected �t and �t lie between the minimum and the maximum buy andsell rates observed along the equilibrium.23Formally, we are assuming that these investors have a set of probability measures over

the possible realizations of �t and �t: Investors compute the minimal expected payo¤sconditional on each possible prior, and decide the investment strategy corresponding tothe maximum of such payo¤s. This idea, which may be thought as an extreme form ofuncertainty aversion, was formalized by Gilboa and Schmeidler (1989).24Indeed, many authors have invoked ambiguity aversion as a possible resolution of

the Equity Premium Puzzle (Chen and Epstein, 2002; Klibano¤, Marinacci and Mukerji,2005).

22

T > ~T ; and in period T + 1 the crash occurs. The smaller the mass of

H-agents, the smaller is the amount of investors who buy stocks at ~T ; and

so the fewer are R-investors with stocks at T and the more are L-investors

with cash at T . Hence, the lower is ST and the higher is BT ; which pushes

towards an higher Tmax: This result is expressed in the following Proposition.

Proposition 7 If z ! 0; there exists a r̂(K;h) < 1 such that Tmax increases

in r for every r � r̂:


5 Discussion

In this Section, we review some of our key ingredients in relation with the

existing literature. We then suggest some policy implications of our results,

and we conclude with some avenues for extensions.

5.1 Related literature

There is a vast literature on speculative bubbles, and we review only some

general themes here.25 Part of the literature builds on the fact that some

information, e.g. relative to the value of fundamentals, is dispersed among

agents. Bubbles are then generated by adding some extra ingredients. Allen,

Morris and Postlewaite (1993), for example, show that bubbles may arise in

a �nite setting with private information only if one introduces also ex-ante

ine¢ ciency, short sale constraints, and lack of common knowledge of agents�

trades. Alternatively, bubbles may occur if agents have subjective (and thus

erroneous) views about how private information is distributed.26

Another stream of literature focuses on the e¤ects of purely mechanical

traders (De Long, Shleifer, Summers and Waldmann, 1990a) or of agents who

form their expectations about future prices simply by extrapolating from past

25For a more detailed review, see Bianchi (2007).26This line of reasoning, however, raises the issue of where subjective priors come from,

and why they survive in equilibrium (see Dekel, Fudenberg and Levine, 2004, for a critiqueof the learning foundations of Nash equilibria with subjective priors).

23

market trends (Cutler, Poterba and Summers, 1990 and De Long, Shleifer,

Summers and Waldmann, 1990b). In a phase of rising prices, however, these

agents would expect the prices to increase with no bounds, and so they

would never understand to be in a bubble nor that the market may crash.

As emphasized in the Introduction, we instead focus on agents with enough

sophistication to understand that they are in a bubble and that the market

must crash.

Moreover, di¤erently from our approach, the literature has typically mod-

eled bubbles and crashes separately. For example, Gennotte and Leland

(1990) focus on the role of hedge funds in provoking the crash while taking as

given the fact that the market is overvalued. Abreu and Brunnermeier (2003)

focus on how coordination issues among rational arbitrageurs may delay the

crash while abstracting from the underlying process generating the bubble.

On the other hand, De Long, Shleifer, Summers and Waldmann (1990b) ex-

plain how feedback trading can generate a bubble but exogenously impose an

end period at which the crash occurs. Scheinkman and Xiong (2003) show

how overcon�dence can sustain speculative trade but do not consider how

the crash may occur.

Finally, by emphasizing cognitive heterogeneity, our work is related to

a wide literature on the limits to information processing.27 In particular,

our idea of equilibrium follows very closely the spirit of Jehiel (2005) who

assumes, in the context of extensive form games, that each player i is char-

acterized by a partition of the set of nodes where other players move, where

each subset of nodes is called analogy class. Player i assesses only the av-

erage behavior of his opponents within each analogy class, and expects this

same average behavior to be played at each node within the analogy class. A

related idea is developed in static games of incomplete information by Eyster

and Rabin (2005).

27See in particular Higgins, 1996 for an exposition of the idea of accessibility in psy-chology and Kahneman, 2003 for economic applications. Many authors have exploredsuch themes in strategic interactions (see Rubinstein, 1998 and the references therein andJehiel, 1995; Jehiel, 2005; Jehiel and Samet, 2007), and �nancial markets (see Hirshleifer,2001).

24

5.2 Information and market e¢ ciency

Our analysis of bubbles focuses on a setting in which information is complete

but some people face limitations in processing all the relevant aspects of such

information. One implication of such analysis is that information availability

per se need not lead to market e¢ ciency. Instead, we point out that informa-

tion accessibility -which focuses on whether information is presented in a way

to ease its interpretation- should matter as well. In this sense, the quest for

market stability may require considering issues of simplicity of information,

or even of information overload, rather than just increasing the amount of

potentially available information.

Along the line of our analysis, one could even argue that some news may

have a destabilizing e¤ect, as they may lead partially sophisticated investors

to get excessively excited, thereby feeding the bubble phenomenon. This is

in a sense what happens within our model when unexpected increases in the

price lead partially sophisticated investors to overestimate the duration of the

bubble and stay invested for too long. If these investors ignored the news,

and in particular the realized price, they would stay invested less long and so

leave less room for bubbles (as shown in the Example above). Information

accessibility and news-driven euphoria may be useful starting points also

for exploring the role of media in stimulating or undermining a speculative

phenomenon.28

5.3 Rational investors and market e¢ ciency

A classic proposition views market e¢ ciency as the result of rational arbi-

trageurs�strategies. However, several models, apart from the present one,

show that rational agents need not have the incentive to immediately stabilize

the market. These include Abreu and Brunnermeier (2003) and De Long et

al. (1990b). These two models di¤er in their predictions. In Abreu and Brun-

nermeier (2003), increasing the share of rational agents reduces the maximal

bubble as it reduces the buying capacity of irrational agents. Conversely, in

28The strong relation between media coverage and abnormal returns has been recentlydocumented e.g. by Dyck and Zingales (2003) and Veldkamp (2006).

25

De Long et al. (1990b), increasing the share of rational agents increases the

size of the bubble as it further distorts irrational agents�expectations.

By contrast, in our model, the relation between the maximal bubble and

the share of rational investors can go both ways, and it depends crucially on

investors�attitudes towards uncertainty. If investors disregard uncertainty,

increasing the share of rational investors makes irrational traders less opti-

mistic and induces rational investors to exit earlier, which reduces the max-

imal bubble (see Proposition 4). Conversely, in a setting with uncertainty

averse agents, speculative investments increase with the share of rational

investors, who face less uncertainty, and such an increase makes irrational

traders more optimistic and it may then allow for longer bubbles (see Section

4.3).

Hence, in our setting, rational investors are not necessarily a stabilizing

force. Instead, market e¢ ciency would be achieved by increasing the frac-

tion of people who admit that their predictions can be imprecise, and that

apparently strange observations may not be the result of chance, but rather

of a wrong model.

5.4 Extensions and future research

While we have described a world with only two cognitive types, a natural

extension would be to enrich the range of cognitive types. There are many

ways this could be done and we will review only a few ideas here.

First, we could consider the case of investors who distinguish a bit between

the various phases of the bubble, thereby further di¤erentiating investors by

how many phases they consider. Apart from generalizing our results, this

exercise might generate additional predictions, for example by revealing that

the order of exit from the speculative market need not be monotonic in

the degree of sophistication (how many phases are distinguished). Perhaps

also, some agents may decide to re-enter the speculative market after hav-

ing exited, creating a richer (and possibly more complicated) set of trading

behaviors.29

29In fact, the threshold property that characterizes optimal strategies in our main analy-

26

Second, we could introduce agents who consider di¤erent aggregate statis-

tics from the data. Instead of average investment strategies, investors could

consider for example average price changes along the bubble, average prices

at the peak of the bubble, average durations of a bubble. For some of these

aggregate statistics, bubbles are less likely to arise.30 We view a more sys-

tematic exploration on which kind of aggregate statistics is likely to give rise

to speculative phenomena as an important direction for future research.

sis (see Lemma 1 and Section 7.1) would hold within each phase that agents distinguish,not necessarily for the entire duration of the market.30For example, if investors only knew the average T in the past bubbles, the bubble

would not arise by standard backward induction arguments. In fact, since our modelis completely deterministic, knowing the duration of past bubbles would be enough toperfectly predict the duration of the current bubble. In a less literal (and more realistic)interpretation of our model, however, each bubble may be somewhat di¤erent from theprevious ones, as described for example by a di¤erent realization of a stochastic element(K, r and z). In such world, having a correct understanding of other investors�strategies,as opposed to simply knowing the realizations of T; would provide a much more usefulinformation about the dynamics of the current bubble.

27

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31

6 Omitted proofs

6.1 Proof of Lemma 1

Notice �rst that if agent i 2 � exits from the market at t; then he must

expect the price to drop at t + 1, i.e. p�;tt+1 < p�;tt , otherwise he would rather

exit at t + 1. By equation (5), and the discussion thereafter, if p�;tt+1 < p�;tt

then K�;tt+2 = 0; so this agent expects the market to close at t+ 2. R-agents�

expectations are correct, so the price indeed drops at t + 1 and the market

closes at t + 2. Hence, these agents will not re-enter at t + 1. Now consider

I-agents. By equation (2), pI;tt+1 < pI;tt is equivalent to BI;tt+1 < SI;tt+1; which,

given equation (11), writes as ��KI;tt+1 < ��(1�K): By equation (5), we have

KI;tt+1 =

��Kt �minf��Kt; ��(1�K)g; so pI;tt+1 < pI;tt is equivalent to

Kt <��(1�K)(1 + ��)

��2:

Now, since Kt cannot increase over time, it must be that

Ks <��(1�K)(1 + ��)

��2for every s � t:

This implies that, at any s � t; I-agents expect the price to drop at s + 1

and the market to close at s+ 2. Hence, such agents will never enter again.

6.2 Proof of Proposition 1

If agent i 2 � expects p�;tt+1 � p�;tt he will buy at t since the strategy of buying

at t and selling at t+1 gives a positive expected pro�t. Notice that, for this

reason, the proposed strategy is optimal even though the agent may be hit by

a liquidity shock which forces him to sell at t+ 1. Conversely, if p�;tt+1 < p�;tt ;

then B�;tt+1 < S�;tt+1. By equation (5), this implies K�;tt+2 = 0; so the agent

expects the market to close at t + 2. Hence, given that the agent expects

that selling at t + 1 would be unpro�table and selling after t + 2 would be

impossible, he does not buy at t. Hence, p�;tt+1 � p�;tt is also necessary for i 2 �

to buy/stay in the market at t. Finally, as already noted in (2), equations

32

(12) and (13) are equivalent.

6.3 Proof of Corollary 1

According to equation (11), condition (13) can be written as

��KI;tt+1 � ��(1�K); (25)

where by (5) we have KI;tt+1 =

��Kt�minf��Kt; ��(1�K)g: If ��Kt < ��(1�K);the agent would expect the price to drop at t and he would exit. This

corresponds to condition (14), since ��(1�K)=�� < W and so ��Kt < ��(1�K)implies Kt < W . If instead ��Kt � ��(1�K); then KI;t

t+1 =��Kt � ��(1�K):

Substituting into (25) gives the result.

6.4 The conditions de�ning T

In order to express conditions (20), (21) and (22) in terms of our exogenous

parameters, notice �rst that, iterating equation (5), the amount of potential

buyers in period s can be written as the di¤erence between the initial amount

of potential buyersK and the accumulated amount of exits up to period s�1,that is

Ks = K �t=s�1Xt=1

[Vt + (1� �t)Kt]: (26)

The volumes of trade induced by the bubble equilibrium are

Vt =

8><>:z(1�K) for t � T � 1;

z(1�K)(1� r) + (1�K)r for t = T;

0 for t = T + 1:

(27)

Hence, using equations (16), (26), (27) and rearranging terms, we get

KT = K � z(1�K)(T � 1);

and

KT+1 = (1� r)[K � z(1�K)T ]� r(1�K):

33

Condition (20) requires �TKT � �T (1�K): With simple algebra, it can bewritten as

T � K � rz(1�K)(1� r) � T1:

We then turn to conditions (21) and (22). To see their structure, we �rst

de�ne the functions

F (T ) � K � z(1�K)(T � 1);

and

G(T ) � (1� r)[K � z(1�K)T ]� r(1�K);

where by construction F (T ) = KT and G(T ) = KT+1: Notice that these

functions are decreasing in T and they both tend to minus in�nity as T goes

to in�nity. Furthermore, with simple algebra, one can show that the function

W (T ); as de�ned in equation (15) and in which �� and �� are given by (18)

and (19), is decreasing and convex in T; and that it tends to 2z(1�K) as Tgoes to in�nity.31 Hence, both F (T ) and G(T ) can intersect W (T ) at most

twice in R+.Suppose indeed that both F (T ) and G(T ) intersect W (T ) twice. Let

T5 and T2 be the roots solving F (T5) = W (T5) and F (T2) = W (T2), with

T5 < T2; and similarly let T4 and T3 be the roots solving G(T4) = W (T4)

and G(T3) =W (T3), with T4 < T3. Since G(T ) < F (T ) for every T; we then

have that T2 > T3 > T4 > T5: In this case, the bubble equilibrium writes as

T 2 [T5; T4) [ (T3; Tmax]; where Tmax � minfT1; T2g:The possibility of two disjoint intervals de�ning the bubble equilibrium

depends on the fact that, in our model, both the amount of potential buyers

at T and I-agents�expectations depend on T , as expressed by the functions

F (T ), G(T ) and W (T ). If F (T ) and G(T ) were constant (i.e. if z were

zero), then we would only have equilibria of the type [T5; T4): According to

condition (21), we would need T � T5 in order to make I-agents� expec-

tations su¢ ciently optimistic and induce them to buy (recall that W (T ) is

decreasing, i.e. I-agents�optimism increases in T ). On the other hand, con-

31Being only algebra, the proof is omitted.

34

dition (22) would require T < T4 : R-agents could not sell too late otherwise

I-agents�expectations would be too optimistic and they would never sell, so

the crash would not occur.

Conversely, if W (T ) were constant, we would only have equilibria of the

type (T3; Tmax]: Condition (21) would require that T � T2: If R-agents sell

too late, I-agents would not buy since the amount of cash observed at that

stage would be too low. On the other hand, condition (22) requires T > T3:

If R-agents sell too early, I-agents would not exit at T + 1, so the crash

would not occur. Hence, it would be optimal to stay in the market rather

than selling at T .

As one expects, equilibria of the type [T5; T4) occur when F (T ) and G(T )

are very high, so the binding constraint is the evolution of I-agents�expec-

tations; while equilibria of the type (T3; Tmax] occur when F (T ) and G(T )

are very low, so the binding constraint is the evolution the amount of cash

observed in the economy. Indeed, for K su¢ ciently high, equilibria of the

type [T5; T4) do not exist, since we have T4 < 1 (as in Example 3.2.1). More

generally, depending on the value of K, r and z, such T2; T3; T4; T5 may not

exist or their value may be less than one. This means that the constraints

de�ned above may or may not bind.

Rather than providing a full treatment of such T2; T3; T4; T5, our analysis

was mainly interested in de�ning conditions for the existence of equilibrium

bubble (as expressed in Proposition 3 and in Section 4.1) and in character-

izing the comparative statics on the maximal equilibrium bubble Tmax (as

expressed in Proposition 4 and in Section 4.3).


Notice �rst that, for every K, z and r; we have T3 < Tmax � minfT1; T2g: Infact, since G(T ) < F (T ) for every T , we have that T3 < T2: Moreover, by

de�nition, G(T1) = 0; so condition (22) holds for sure at T1 and then T3 < T1:

Given the shape of the function W (T ) described in Section 6.4, the bubble

equilibrium exists if and only ifW (T ) and F (T ) intersect at least once, i.e. if

there exists a T2 � 1 such that F (T2) =W (T2): In fact, when this is the case,

35

Tmax can always be sustained as equilibrium. Hence, a su¢ cient condition

for the existence of a bubble equilibrium is that W (T ) and F (T ) intersect

once and only once, that is the case when K � W (1): With some algebra,

one writes

K � W (1)() K � [z(1�K)(1� r) + (1�K)(1 + r)](3� r)(1� r)2 : (28)

Condition (28) can be rearranged to de�ne a K� such that if K � K� then

K � W (1); and so a bubble equilibrium exists. Moreover, one can see that

such K� is always smaller than one, and it increases in r and z.32

6.6 The conditions de�ning Tmax

We now turn to the analysis of the conditions under which T1 or T2 de�nes

Tmax � minfT1; T2g: Notice �rst that T1 < T2 if and only if W (T1) < F (T1):By de�nition of T1; F (T1)(1� r) = ST and ST = z(1�K)(1� r)+ r(1�K);so W (T1) < F (T1) writes

z(1�K)(1� r) + r(1�K)1� r > W (T1): (29)

In Section 3.3, we claimed that Tmax = T2 when r is small; and Tmax = T1when z or K are small: We now show that this is indeed the case. Consider

the �rst claim. Rearranging condition (29), we can de�ne a threshold �r such

that T1 < T2 if and only if r > �r: Such threshold is implicitly de�ned by

�r = P (�r); where

P (r) � W (T1)� z(1�K)W (T1) + (1� z)(1�K)

: (30)

In fact, P (r) is increasing inW (T1); andW (T1) is increasing in r. Moreover,

P (0) > 0 and P (1) < 1: Hence r > P (r) holds for r > �r; where �r is uniquely

de�ned by �r = P (�r):

Now consider the case of z ! 0; i.e. the probability of liquidity shocks is

very small. Both T1 and T2 tend to in�nity as z tends to zero, but T2 exceeds

32Equation (28) can alternatively be rearranged to de�ne a r� and a z� such that ifr � r� or if z � z�, then a bubble equilibrium exists.

36

T1. In fact if z ! 0; then z(1 �K) ! 0; T1 ! 1 and W (T1) ! 0: Hence,

P (r)! 0; so r always exceeds P (r) and Tmax = T1.

Finally, consider the conditions on K: Condition (29) can be rearranged

as

K <r + (1� r)[z �W (T1)]

z(1� r) + r � Q(K):

Notice �rst that if K = 1; then W (T1) = 0 and so Q(1) = 1: That is, if

K = 1; then T1 = T2. For K = 0, no bubble equilibrium exists, so we only

have to consider K � Kmin, where Kmin corresponds to the case T1 = 1 and

it writes as

Kmin �r + z(1� r)1 + z(1� r) :

Now, it can be shown (with simple algebra) that Q(Kmin) > Kmin; which

means that T1 < T2 for K = Kmin:


By di¤erentiating equation T1 = (K � r)=[z(1�K)(1� r)], one sees that T1increases in K and decreases with z and r: To see the e¤ects on T2, de�ne the

function L(T ) � F (T )�W (T ): By de�nition, L(T2) � 0: Di¤erentiating thefunction L(T ); one can see that it decreases in T2; z and r and it increases in

K: Hence, by the implicit function theorem, T2 increases in K and decreases

with z and r. The second part of the Proposition can be shown by noticing

that if K ! 1 or z ! 0; then z(1�K)! 0: Both T1 and T2 tend to in�nity

as z(1�K)! 0:


Condition (24) writes ��KT+1 < ��(1�K). Recall that condition (21) requires��KI;T

T+1 � ��(1�K): Hence, conditions (24) and (21) jointly require KT+1 <

KI;TT+1: Recall that KT+1 = �TKT �ST , and KI;T

T+1 =��KT � ��(1�K): Hence,

KT+1 < KI;TT+1 if and only if

(�T � ��)KT + [��(1�K)� ST ] < 0: (31)

37

Consider the �rst term in (31). Recall that �T = (1�r) and �� = (T�r)=(T+1); so �T < �� requires (T +1)(1� r) < (T � r); that is rT > 1: Now considerthe second term in equation (31). Recall that �� = ((T � r)z+1+ r)=(T +1)and ST = r(1 � K) + z(1 � K)(1 � r): Hence, ��(1 � K) < ST requires

r(1�K)� z(1�K)Tr > 1�K� z(1�K), that is rT > 1: Hence, condition(31) is satis�ed if and only if rT > 1: In particular, recall that we must have

T � T1; where T1 = (K � r)=[z(1 � K)(1 � r)]; so condition (31) requiresr > [z(1 � K)(1 � r)]=(K � r): Doing the algebra, the last inequality issatis�ed for r 2 (r1; r2); where r1 > 0. Hence, there exists a rmin > r1 > 0such that if r � rmin condition (31) cannot hold:


Notice �rst that, as in equation (21), L-investors buy/stay in at t if and only

if Kt � W; while H-investors buy/stay in at t if and only if Kt � W (");

where

W (") � (1�K)(�� + ")(1 + �� ")�� "

�2 :

Since W (") increases in "; H-investors always sell before L-investors.33 We

then de�ne an equilibrium in which for t < ~T no investor wants to exit

(apart from the liquidity traders); H-investors leave the market at ~T , selling

to rational agents R and to low ambiguity averse agents L:34 At T > ~T ;

rational investors sell to L-agents. At T + 1; L-agents realize that the crash

is about to occur and they want to sell, while no one is willing to buy. The

33W (") is simply obtained from (21) by replacing �� with �� "; and �� with �� + ".34Since the distribution of reservation prices, cognitive type and probability of liquidity

shock are all independent, the proportion of R and L in the speculative market remainsconstant until T . Hence, for t 2 ( ~T ; T ); the proportion of R-investors in the market is

r

r + (1� h)(1� r) ;

and, similarly, the proportion of L-agents is

(1� r)(1� h)r + (1� h)(1� r) :

38

crash occurs and the market closes.

We are interested in how Tmax varies with r; for a given proportion of

ambiguity averse agents h.35 Consider the e¤ects on T1; recalling that T1is de�ned by BT = ST : In our equilibrium, ST includes all R-investors with

stocks at T and the exogenous sales z(1�K), whileBT includes all L-investorswith cash at T . That is, ST = �t(1�K); where

�t = z + (1� z)r

1� h(1� r) ;

and BT = �tKT ; where

�t =(1� r)(1� h)1� h(1� r) ;

and

KT = K� z(1�K)(T �1)� (1� r)h[(1� z)(1�K)+K� z(1�K)( ~T �1)]:

In order to simplify the exposition, from now on we assume that H-investors

perceive enough uncertainty to be induced to sell immediately, i.e. that " is

large enough to have ~T = 1:36 Hence, BT � ST de�nes the condition T � T1;where

T1 �1

z(1�K) [K + z(1�K)� (1� z + zK)(1� r)h

� z(1�K)� hz(1�K)(1� r)� (1� z)(1�K)r(1� r)(1� h) ]: (32)

After simple algebra, we can see that

@T1@r

=1

z(1�K) [h(1� z + zK)�1�K

(1� r)2(1� h) ]; (33)

35The e¤ect of h is clear: Tmax decreases with h; that is the fraction of people leavingthe market early.36For example H-investors may think that �t and �t are respectively drawn by dis-

tributions with mean �� and �� and support on [0; 1]. As they are extremely ambiguityadverse, they assume �t = 0 and �t = 1 for all t, so they exit as soon as possible. In otherwords, given that there is a one-to-one mapping between ~T and "; we now consider ~T asan exogenous parameter of the model.

39

which is positive when

r � 1�

s1�K

h(1� h)(1� z + zK) : (34)

Proposition 7 claims that if z ! 0, then Tmax increases in r for every r � r̂;where

r̂ � 1�

s1�Kh(1� h) :

To see that, notice �rst that Tmax = T1 when z ! 0: In fact, one can replicate

the analysis of Section 6.6 in the setting with ambiguity aversion and write

that T1 < T2 if and only if r exceeds a threshold implicitly de�ned by

r >(1� h)[W (T1)� z(1�K)]

(1� h)[W (T1)� z(1�K)] + 1�K:

If z ! 0 the right hand side of the last equation tends to zero, and so T1 < T2:

Substituting z = 0 into equation (34) gives the result.

40

bubbles and crashes with partially sophisticated...

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